Making use of graphene's valleys

Graphene has two distinct valleys in its electronic structure, in which the electrons have the same energy. Theorists have predicted that creating an asymmetry between the two valleys will coax graphene into exhibiting the so-called valley Hall effect (VHE). In this effect, electrons from the two valleys move across the sample in opposite directions when the experimenters run current along the sample. Gorbachev et al. achieved this asymmetry by aligning graphene with an underlying layer of hexagonalboron nitride (hBN) (see the Perspective by Lundeberg and Folk). The authors measured the transport characteristics of the sample, which were consistent with the theoretical predictions for the VHE. The method may in the future lead to information processing using graphene's valleys.

Abstract

Topological materials may exhibit Hall-like currents flowing transversely to the applied electric field even in the absence of a magnetic field. In graphene superlattices, which have broken inversion symmetry, topological currents originating from graphene’s two valleys are predicted to flow in opposite directions and combine to produce long-range charge neutral flow. We observed this effect as a nonlocal voltage at zero magnetic field in a narrow energy range near Dirac points at distances as large as several micrometers away from the nominal current path. Locally, topological currents are comparable in strength with the applied current, indicating large valley-Hall angles. The long-range character of topological currents and their transistor-like control by means of gate voltage can be exploited for information processing based on valley degrees of freedom.

Berry curvature is a physical field intrinsic to some crystal lattices, which can dramatically affect the transport properties of materials (1–6). Topological effects, although known for some time (7), have gained attention recently in connection with the discovery of topological insulators (8–11). In these materials, topological bands lead to new phenomena such as topologically protected edge-state transport in zero magnetic field (12–14). No less striking, however, is the expected impact of Berry curvature on bulk transport, leading to topological currents flowing perpendicular to the applied electric field E (5, 6). The nondissipative nature of these currents, ensured by their transverse character, resembles that of Hall currents. However, topological currents can arise in the absence of magnetic field B and even without breaking time-reversal symmetry (TRS). In contrast to cyclotron orbits in a magnetic field drifting perpendicular to E, topological currents originate from perfectly straight trajectories that skew left or right relative to E (Fig. 1A). In materials whose electronic structure has more than one valley (4, 15, 16), topological currents in different valleys have opposite signs and, if intervalley scattering is weak, can add up to produce long-range topological charge-neutral currents.

Graphene placed on top of hexagonal boron nitride (G/hBN) (Fig. 1) affords an exceptional venue for inducing and manipulating topological bulk currents at B = 0 for the following reasons. First, graphene’s band structure possesses a nonzero Berry’s phase (17), which is a prerequisite for the existence of Berry curvature. Second, if crystallographic axes of graphene and hBN are aligned, global A/B sublattice asymmetry is introduced (18–20). This gives rise to broken inversion symmetry and a finite Berry curvature. Third, the high electronic quality of graphene protects topological currents against intervalley scattering, allowing them to propagate away from the applied current path. The valley-Hall effect (VHE) creates an electrical response in remote regions, which can be exploited to detect the presence of topological currents experimentally. The nonlocal response is expected whenever the Fermi level in graphene is tuned into one of the Berry curvature hot spots (Fig. 1B). This approach, besides offering a direct probe of topological currents, provides a precision tool for mapping out the Berry curvature in G/hBN Bloch bands. So far, it has proven challenging to probe bulk topological currents without applying B. In particular, this was the case for the anomalous Hall effect in magnetic metals, which was previously perhaps the cleanest system available by which to study topological bands (6).

Experimentally, we studied G/hBN superlattices (in which the graphene and hBN lattices were aligned) fabricated following the procedures described in (21). Fifteen superlattice devices similar to the one shown in the inset of Fig. 1B were investigated. The charge carrier mobilities μ varied from 40,000 to 80,000 cm2 V−1 s−1, and the longitudinal and Hall resistivities (ρxx and ρxy, respectively) exhibited essentially the same behavior as reported previously (18, 19, 21, 22). Namely, pronounced peaks in ρxx were observed at the main neutrality point (NP) and at carrier density n ≈ ±3 × 1012 cm−2 (Fig. 1B and fig. S1). The peaks in ρxx were accompanied by a sign reversal of ρxy indicating the emergence of new NPs in the valence and conduction bands of graphene (18–22). Both encapsulated and nonencapsulated structures were investigated, with the former having an additional hBN crystal placed on top of a G/hBN superlattice to protect it from the environment (20, 21). Some of the latter showed activation behavior at the main NP (fig. S2), yielding a bandgap 2∆ of 350 ± 40 K (20), which is in agreement with earlier transport (18, 19) and spectroscopy (23) measurements. Our encapsulated devices, despite higher electronic quality, exhibited no activation behavior at the main NP, with the ρxx value saturating at <10 kilohm for temperatures T below 50 K (19, 21). Although this behavior remains to be understood, it is likely that devices that do not show activation behavior in transport properties still have a bandgap (20). In this regard, observing the activation behavior usually relies on midgap impurity states that pin the Fermi level inside the gap (20). However, few midgap states are expected in high-quality G/hBN devices. Alternatively, spatial charge inhomogeneity can “short-circuit” the activation behavior by allowing current to circumnavigate the insulating regions. This would also obscure the activation behavior, leading to metallic-like transport in ρxx. However, charge inhomogeneity is expected to have relatively little effect on the VHE; being a non–sign-changing function of density (Fig. 1B, inset), it should not average out. Therefore, the activation behavior is not essential for observing topological currents (20). Indeed, both encapsulated and nonencapsulated graphene were found to exhibit very similar nonlocal response.

The central result of our study is shown in Fig. 1B. The nonlocal resistance RNL was determined by applying current between, for example, contacts 4 and 8 and measuring voltage between contacts 3 and 9 (Fig. 1B, micrograph). RNL exhibits large sharp peaks at the main and hole-side NPs (unless stated otherwise, all the presented data refer to zero B). A striking feature of the observed nonlocality is its narrow range in n (Figs. 1B and 2A). Unlike ρxx, which follows the 1/n dependence typical for graphene and remains sizeable (>100 ohm) over the entire range of accessible n (Figs. 1 and 2 and fig. S1), RNL decays rapidly with n and completely disappears under noise for densities >1011 cm−2 away from the NPs. The dependence can be approximately described by , with α ≈ 2.5 to 3 (Fig. 2B). A nonlocal voltage can also appear because of stray charge currents, described by the van der Pauw relation RNL ~ ρxx exp(–πL/w), where L is the distance between the current path and voltage probes and w is the device width (Fig. 1B, micrograph) (24). For typical aspect ratios L/w ≈ 4, the formula yields ≈0.01 ohm. The magnitude of this contribution and its n dependence, which follows ρxx(n), are clearly incompatible with the observed nonlocal response.

(A) Behavior of ρxx and RNL near the main NP in superlattice devices (green and red, respectively). The blue curve is for a nonaligned, reference device (≈10° misalignment between graphene and hBN crystal axes). L ≈ 3.5 μm; w ≈ 1 μm; T = 20 K. (B) The same data as (A) on a logarithmic scale. Away from the NP, ρxx exhibits the conventional 1/n dependence (green symbols) and can be described by ρxx(n) ∝ 1/(n2 + δn2)1/2, where δn ≈ 1.5 × 1010 cm−2 accounts for charge inhomogeneity (light purple line). Measured RNL (red symbols) and the Berry curvature model (black curve, Eq. 3). Model uses the above δn and bandgap 2 Δ = 360 K found from local measurements (18–20). (C) Nonlocal signal decays exponentially with increasing L. Color refers to similar devices with the same L or the same device with two different L; w ≈ 1 μm, 20 K; 10 devices in total. The black star is the anomalous contribution to ρxx observed in the bend geometry, which is consistent with strong topological currents induced locally (20). (Inset) RNL remains sizeable at L ≈ 7 μm.

We have also investigated how RNL depends on L and found an exponential dependence exp(–L/ξ) with ξ ≈1.0 μm (Fig. 2C). This ξ value is close to w and much larger than the elastic mean free path of ≈0.1 μm estimated from μ for the range of n where RNL appears. Furthermore, RNL exponentially decreases with increasing temperature T so that no nonlocal signal is detected above 150 K, whereas ρxx remains large at this T at the main NP (figs. S2 and S3). Nonencapsulated devices exhibited practically the same behavior of RNL as a function of n and T (fig. S4), but the absolute value of RNL was somewhat smaller than that in encapsulated devices (20). In addition, we investigated the effect of charge inhomogeneity δn on RNL, which was controlled by sweeping to progressively larger gate voltages Vg beyond the hole-side NP (20). When δn was increased by a factor of ≈2, RNL could sometimes change by more than an order of magnitude (20). The inhomogeneity enhanced RNL at the main NP and suppressed it at the hole-side NP (fig. S5). The difference is attributed to a narrow energy width of the secondary Dirac spectrum (20).

To confirm the key role played by Berry curvature, we verified that the nonlocal response was absent in G/hBN systems without alignment and, accordingly, with no detectable superlattice effects (18–22). This is illustrated in Fig. 2A, which shows RNL for aligned and nonaligned devices with the same μ and in the same geometry. In the nonaligned devices (>20 measured), no nonlocal signal could be observed even at our maximum resolution of ∼0.1 ohm (Fig. 2A, blue curve). Therefore, the observed RNL cannot be explained by charge-neutral flow of spin and/or energy, which are indifferent to crystal alignment. The latter flows also require broken TRS and completely disappear in zero B, as reported previously (24–26). In our superlattice devices, a contribution of spin/energy flows becomes appreciable only for B > 0.1 T (fig. S6), leading to rapid broadening of the RNL peaks, which is in agreement with (24).

We have also considered the possibility that the observed nonlocality at B = 0 may originate from an edge transport mechanism. For example, topological materials can support gapless edge modes that coexist with the gapped bulk. Such modes could in principle mediate nonlocal charge transport. However, our experiments provide no evidence for metallic conductivity along the device edges. First, the measured nonlocal response was similar for all our devices, independent of whether they exhibited metallic or insulating behavior in ρxx. Second, atomic force microscopy studies show that the moiré pattern associated with the alignment extends all the way to the device edges, and there are no distinct edge regions (19). Third, extrapolating the dependence in Fig. 2C to small L, we found that the edge transport scenario would require metallic conductivity along edges of ≈2e2/h over distances of L > 1 μm. This would imply perfect edge-state transport, which is unlikely at such length scales in the presence of strong intervalley scattering expected at microscopically rough edges.

Proceeding with the analysis, the aligned superlattices are comprised of hexagonal unit cells, each representing a commensurate graphene/hBN region of ∼10 nm in size, which is surrounded by a strained boundary (19). All unit cells are characterized by A/B sublattice asymmetry of the same sign, giving a preferred chirality over the entire structure (19, 20, 25). For our typical n < 1011 cm−2, Fermi wavelengths are larger than 100 nm, which exceeds the superlattice periodicity by a factor of 10. This large wavelength/period ratio renders contributions from spatially varying couplings insubstantial.

We will show below that the observed nonlocality features are consistent with bulk topological currents expected for a gapped Dirac spectrum. The mechanism by which Berry curvature generates topological currents can be elucidated by the semiclassical equations of motion (5)(1)where Ω is the Berry curvature density and v(k) is the group velocity of Bloch electrons. The Lorentz force term ev × B describes the conventional Hall effect. Berry curvature gives rise to an “anomalous velocity,” k × Ω(k), which is of the same structure as the Lorentz term but in momentum space and leads to transverse currents (Fig. 1A). Such currents may appear in zero B [without breaking TRS that requires Ω(k) = −Ω(−k)] as long as inversion symmetry is broken, Ω(k) ≠ Ω(−k), as is the case of our aligned devices with globally broken A/B sublattice symmetry (19). For B = 0, transverse currents arise solely from Ω(k), yielding Hall-like conductivity σxy given by the sum of Berry fluxes for all occupied states in the Fermi sea (3)(2)with f(k) being the Fermi function and the factor of 2 accounting for spin degeneracy. Because Berry curvature is odd in energy (20), σxy has the same sign for both electrons and holes. This contrasts with the conventional Hall conductivity that is sign-changing under carrier-type reversal. As a result, σxy given by Eq. 2 is less susceptible to smearing by inhomogeneity. This mechanism yields a nonzero RNL whenever the Fermi level is tuned through Berry curvature hot spots. Their extent in energy is given by half the bandgap Δ ≈ 180 K, which translates into n ≈ 2 × 1010 cm−2 and agrees well with the ultranarrow width of our RNL peaks.

Because of TRS, the electric field generates topological currents (Eq. 1) with opposite transverse components in graphene’s two valleys, K and K′ (Fig. 1A), to create the charge-neutral VHE, , where . As illustrated in the inset to Fig. 1B, topological currents can result in a VHE conductivity of ≈2e2/h. In the absence of intervalley scattering, the charge-neutral currents can persist over extended distances and mediate nonlocal electrical signals (24–28). The resulting nonlocal resistance RNL can be understood as originating from the VHE and a reverse VHE (20), by analogy with nonlocal transport mediated by charge-neutral spin or energy flow (24–28). Yet unlike the latter, the VHE-induced nonlocality appears without TRS breaking—that is, at zero B. This behavior, as well as the narrow range of n over which RNL is observed, is a telltale sign of bulk topological currents. The analysis outlined above yields the model expression (20)(3)The peak in RNL(n) can be described by Eq. 3 with no fitting parameters (Fig. 2B).

The measured spatial decay with ξ ≈ 1.0 μm is consistent with intervalley scattering occurring at graphene edges and/or at atomic-scale defects (20). The large values of RNL at L of several micrometers also imply extremely strong topological currents locally, within the path of the applied current. By extrapolating the observed L dependence to L < 1 μm, Fig. 2C yields RNL ∼10 kilohm. According to Eq. 3, this translates into and order-one Hall angles, which is in agreement with the VHE expected for weak intervalley scattering. Furthermore, similar to classical magnetotransport, changes in the direction of current flow can lead to additional resistivity. For σxyρxx ∼ 1, the classical magnetoresistance reaches a value of ∼ρxx when carriers of opposite sign are involved. A valley analog of this extra resistance may explain anomalous contributions of ∼10 kilohm in ρxx, which are observed at short distances L ≈ w by using the bend geometry (fig. S7). Parenthetically, the intrinsic VHE mechanism discussed above, which provides excellent agreement with our experimental results, may coexist with extrinsic VHE mechanisms such as skew scattering and side jumps (6). Although their role in graphene superlattices remains to be examined, such mechanisms also originate from Berry curvature and arise under the same symmetry conditions as the intrinsic contribution.

Last, sharp changes in RNL with Vg (130-nm-thick dielectric) (Figs. 1 and 2) amount to a transistorlike response with a slope of ≈100 mV/dec—that is, the detected voltage changes by a factor of 10 by varying Vg by ≈100 mV. Although the peaks in RNL broaden with increasing T and disappear above 100 K (because of the relatively small Δ), one can envision electronic devices based on the valley degrees of freedom (29), which would become practical if larger bandgap values are achieved. To explore this further, we fabricated a superlattice device with a short top gate (15-nm dielectric) placed between the current and voltage contacts used for nonlocal measurements (fig. S8). Valley currents in this case could be switched on and off, similar to the case of a field effect transistor, by a gate voltage of ≈10 mV at 20 K (20). It is feasible to further reduce the thickness of the top gate hBN dielectric down to 2 nm, which would translate into a gate response down to <2 mV/dec at this T. Although further analysis is necessary, these results may indicate that subthreshold slopes better than those achievable for conventional charge-based processing devices (30) are possible.

Acknowledgments: This work was supported by the European Research Council, the Royal Society the Office of Naval Research, the Air Force Office of Scientific Research, the Engineering and Physical Sciences Research Council (UK), and the National Science Foundation.